The left-back made the game safe after a brace from the mercurial Lionel Messi and a goal from David Villa helped the Catalan club overturn a 2-0 deficit from the first leg of their last-16 tie.
Milan's success at the San Siro left many criticising some of Barcelona's stars for failing to produce the goods on Europe's grandest stage.
But their emphatic 4-0 triumph at the Nou Camp answered many questions, with Alba urging his side to continue that form into the weekend when they return to Primera Division action at home against Rayo Vallecano on Sunday.
"Outside of the club there was a lot of doubt, but we were all calm," he told a press conference.
"We weren't playing badly in these three games against Milan and Real Madrid but we just lacked the same intensity and aggression. The other day we saw the real Barcelona and now we have to maintain this level.
"We have to continue this level of play and aggression until the end of the season."
Of his goal, Alba added: "It was one of the most important goals of my career because it sealed the win after we'd suffered a lot in the last 10 minutes".
While much focus centred on Friday's Champions League draw, with Barca paired with Paris St Germain in the last eight, Alba is very well aware they can not afford to take their eyes off the challenge posed by Vallecano.
Barca are 13 points clear at the top of La Liga, but Alba insists the players are not content to rest on their laurels against a Vallecano side that lie eighth in the table but have won just four of 13 league matches on their travels.
"We have to try and win the league as soon as possible," he said.
"Rayo are having a magnificent season. They like to play with the ball and they are aggressive without it. I like the way they play."
Stand-in head coach Jordi Roura is likely to be without midfielder Xavi, who suffered an injured thigh against Milan, while goalkeeper Victor Valdes will sit out due to suspension.
Messi hit his 52nd and 53rd goals of the season on Tuesday, but Vallecano midfielder Roberto Trashorras says the side have not decided on a specific way to derail the Argentinian.
"We have not spoken especially about Messi," the former Barca man told Vallecano's official website.
"We have talked about how to disrupt their rhythm and we intend to complicate the game for Barcelona."
Defender Jose Manuel Casado, midfielder Javi Fuego and striker Leo Baptistao will be unavailable for the trip to the Nou Camp against a club goalkeeper David Cobeno says Vallecano need to be at their very best to beat.
"Of course we will be motivated against Barca," he said.
"We're playing against one of the best teams in the world in a week where they have played and won a very important game.
"They will be very motivated and we have to try to do what we've done all year: to enjoy and look for something positive.
"Even when they are at 80 per cent they are better than other teams. We have to be at 120 per cent in order to get something and if they are lower because they are more relaxed, so much the better for us."